The ways to destroy a Horcrux are limited and complex, with the objects being durable and resistant to most physical damage and magical spells. In the unlikely circumstance that an average wizard comes across one of the objects, it's doubtful they'd carry the knowledge or weapons needed to destroy a Horcrux entirely. Only the most potent forms of destructive magic and processes have the capability of complete destruction, especially if the creator had enchanted the objects on top of their already established self-defense mechanisms to ensure they remain unharmed.
The first thing to remember regarding the destruction of Horcruxes is that they must be thoroughly destroyed for their powers to be nullified. Horcruxes can be repaired physically or magically if they're not damaged enough. In the case of a living being that was created into a Horcrux, the only way to destroy them is by killing them. The destruction of all the Horcruxes that a wizard makes will result in their anchor to the living world being gone, meaning they'll be rendered mortal and, therefore, able to be killed again. Only four documented methods were strong enough to destroy a Horcrux: venom from a Basilisk, Fiendfyre, the Killing Curse, and by the creator. Horcruxes may appear to bleed when they're destroyed, as seen with the diary bleeding ink and the diadem oozing a blood-like liquid. Once a Horcrux is destroyed, it's possible that the creator can feel or sense its destruction, but nothing has ever been confirmed on the matter.
Basilisk venom is a dangerous substance to acquire, having highly corrosive properties. Venomous snakes like a Basilisk have a venom sac in their heads where the venom is stored until it's pushed through a duct and into the hollow fangs, exiting through a hole at the bottom of the fang and into its target. It's highly potent and can kill someone within about a minute after being bit, causing drowsiness and blurry vision before death. If an individual managed to gather some of the venom, it's known to be very long-lasting, as proven by Hermione Granger being able to use the venom in a Basilisk fang from a beast that was dead for almost five years. The Sword of Gryffindor managed to become infused with the venom after Harry Potter killed the Basilisk in the Chamber of Secrets, possible only because goblin-made items absorb things that only make it stronger, making the sword an effective tool to destroy Horcruxes. There's only one known antidote to Basilisk venom: phoenix tears. Phoenix tears are rare to obtain and have spectacular healing abilities, going as far as saving someone from the brink of death, similar to the effect of drinking unicorn blood without having a cursed life afterward.
Fiendfyre, having the spell incantation of "Pestis Incendium," is a dark magic curse that produces enchanted, destructive flames. Its flames are vast and hot, destroying everything in its path. These fast-acting flames ignite with a roaring, billowing noise and can take the form of beasts in their flames. They can become semi-sentient, having a desire to burn everything they can. They're notorious for being hard to control, even by experienced wizards, and spread rapidly. It's unknown if these flames can be extinguished by burning out on their own, as they cannot be extinguished by regular water, even if water-making spells conjure the water. The only known way to extinguish its flames is with the counter-curse for Fiendfyre, though its incantation is unknown and has some difficulty in its casting. Combined with all of these aspects, it makes for a great, though mostly uncontrollable, way to destroy Horcruxes.
The Killing Curse has the incantation of "Avada Kedavra," the dark magic curse that causes a painless and instant death to a living target when cast successfully. The casting of the curse causes a rushing sound and green light from the caster's wand. When used on a living Horcrux, the curse seemingly targets the soul fragment that was placed within the target rather than the target itself. However, it has only been attempted on Harry Potter, who had previously gained the protection of sacrificial love from his mother's death and the administration of phoenix tears after the Basilisk bit him in the Chamber of Secrets. It's unknown if it would work the same way on another animate Horcrux that doesn't have those protections, like Nagini, with the possibility that the curse would kill the target instead of just the foreign soul fragment. It may be possible to destroy inanimate Horcruxes like the Ravenclaw diadem with this method if the curse targets the soul fragments, but it's never been seen before, remaining a mystery for now.
The most unheard-of way to destroy a Horcrux is by the creators doing it themselves. The book containing the most information about Horcruxes, Secrets of the Darkest Art, has written a way to reunite the soul fragments scattered across one or more Horcruxes, which would essentially destroy them. The way to achieve this is by the creator having genuine remorse for what the creator had done to create them. This method is rarely seen in the wizarding world, as finding a dark wizard willing to undo their commitment to immortality and completion of the harsh process of making a Horcrux is extraordinarily improbable. Even if a dark wizard met the criteria, the process has been known to be excruciatingly painful and possibly fatal, which would deter many from attempting.
The evening of May 29th, 1993, marks the day Voldemort's first Horcrux was destroyed. Once the Chamber of Secrets was opened, Harry Potter, Ron Weasley, and Gilderoy Lockhart went in, where Lockhart never intended to fight the Basilisk, instead using Ron's broken wand to attack both boys, resulting in his memory getting wiped. Harry proceeded further, finding an unconscious Ginny Weasley and the diary, which he didn't know was a Horcrux then. The materialized figure fueled by the diary's soul fragment would reveal his true identity and plan to Harry, unleashing the Basilisk. Harry fought against the beast with the assistance of Dumbledore's phoenix, Fawkes, and with the help of the Sword of Gryffindor, which materialized from the Sorting Hat. As Harry slays the beast by stabbing the Basilisk through the roof of its mouth with the sword, he accidentally infuses it with venom and gets bit. Once the Basilisk was down, Fawkes healed Harry's wound with tears, saving his life, and Harry took the beast's fang and stabbed into the diary with it repeatedly until it was irreparable, destroying it as a Horcrux.
Marvolo Gaunt's ring was destroyed in July 1996, the second Horcrux to be destroyed. Dumbledore continued looking into possible Horcruxes of Voldemort and concluded that one could be hidden away in the Gaunt Shack since the location was important to Voldemort's past. Dumbledore traveled to Little Hangleton and discovered the shack hidden by weeds and brush. He got past the enchantments protecting the location and investigated, finding a golden box underneath the floorboards that held the ring. He had brought the Sword of Gryffindor with him to destroy the Horcrux, but upon recognizing the stone within the ring as the Resurrection Stone, part of the Deathly Hallows, he put the ring on. This then triggered the deadly curse enchanted into the ring, and it started to spread through Dumbledore's body. After returning to Hogwarts, Dumbledore finally cracked the stone open with the sword, destroying it as a Horcrux but not ending the curse.
Albus Dumbledore tasked Harry Potter with a Horcrux hunt in June 1997 to find and destroy the remainder of Voldemort's Horcruxes. Harry, Ron, and Hermione had managed to acquire Slytherin's locket. However, they were forced to keep it safe for an extended period, taking turns wearing it, which resulted in the Horcrux gaining power and affecting them. The trio couldn't destroy the locket until December 28th, 1997. The trio had received news that some students were in trouble for trying to steal the Sword of Gryffindor at Hogwarts, one of which being Ginny, leaving Ron particularly angry from the locket's effect, which resulted in him storming off. While in the Forest of Dean, Harry would see a bright white light while he was on guard duty, which turned out the be Severus Snape's Patronus, and followed it to a frozen pond, where the Sword of Gryffindor was placed. Harry tried to get the sword but ultimately came to the conclusion that he had to get into the pond to retrieve it. While in the pond to gather the sword, the locket had other plans, strangling him, then keeping him underwater and nearly drowning him. Ron returned that moment, grabbing Harry, ripping the locket off him, and grabbing the sword. After some reconciliation, it was decided that Ron should destroy the locket since he had retrieved the sword. Harry opened the locket using Parseltongue, saying "open" to the locket, which revealed images of Voldemort's eyes before they had turned red, which prompted the locket to torture Ron with his greatest fears, all while Harry was yelling at him to snap him out of it. Eventually, he succeeds in snapping out of it and smashes both glass windows of the locket with the sword, destroying it as a Horcrux.
The day of the Battle of Hogwarts, May 2nd, 1998, is when the remainder and majority of Voldemort's Horcruxes were destroyed. The golden trio would come to theorize that Hufflepuff's cup was located in Bellatrix Lestrange's vault at Gringotts. After they steal the cup from the vault, this angers Voldemort as he realizes his secret has been discovered, resulting in him gathering an army and his Death Eaters to launch a final attack at Hogwarts. During the battle, Hermione and Ron went to the Chamber of Secrets thanks to Ron's passable Parseltongue, and they gathered some Basilisk fangs. Hermione used one of these fangs to destroy the cup, as the venom was still potent enough to destroy a Horcrux.
Hermione and Ron then regrouped with Harry on the way to the Room of Requirement, informing him of the cup being destroyed by Hermione. Once they entered, they split up to cover more ground in search of Ravenclaw's diadem. Harry was ultimately the one who found the Horcrux, but he got ambushed by Draco Malfoy, Vincent Crabbe, and Gregory Goyle before he could gather it. They had a brief altercation, where Crabbe and Goyle had intentions of killing, resulting in Crabbe using Fiendfyre. Crabbe couldn't control the flames of this dark curse, as the trio would attempt to escape after they found two broomsticks, with Harry taking one and Ron and Hermione taking the other. As they flee, Harry goes to save Malfoy, grabs the diadem as it falls, and then escapes the room. After finding safety, they examined the diadem, which the fire of Fiendfyre had destroyed and, therefore, destroyed as a Horcrux.
During a one-hour ceasefire during the battle, Harry uncovered Snape's memories in the Shrieking Shack after Nagini had bitten Snape and injected him with venom. The dying man's memories revealed that Harry had unintentionally been turned into a Horcrux, meaning he had to let Voldemort kill him to be able to cause Voldemort's downfall. With this knowledge, Harry entrusted Neville Longbottom with killing Nagini and followed two Death Eaters to their camp. Once at the camp, Harry has a confrontation with Voldemort, where he is willing to let Voldemort use the Killing Curse on him to kill him. Harry survived due to the protection of sacrificial love, which Voldemort unknowingly helped preserve when he drank the regeneration potion containing Harry's blood. It acted somewhat like a Horcrux because it helped anchor the protection via another being. The Killing Curse only targeted the soul fragment of Voldemort within Harry, meaning he was no longer a Horcrux. Voldemort didn't know Harry survived, as Harry had managed to fake his death.
The Death Eaters were fueled by Harry's supposed death, marching to the castle and demanding surrender. Nagini was the only remaining Horcrux, which Neville was tasked with killing. As it was believed Harry had died, Neville lunged at Voldemort, refusing to join the dark wizard, resulting in Voldemort casting the Full Body-Bind Curse on him, placing the Sorting Hat on his head, then setting it on fire. Chaos would quickly follow, with creatures from the Forbidden Forest coming out and attacking the Death Eaters, allowing Neville to free himself. Neville was able to summon the Sword of Gryffindor from the Sorting Hat and used it to slay Nagini, putting an end to all of Voldemort's Horcruxes and leaving him susceptible to death.